Preheat the oven to 375°F (190°C). Lightly grease a 9 × 13-inch baking dish. If the chicken breasts are very thick, slice them horizontally into thinner pieces so they cook evenly. Season both sides with garlic powder, Italian seasoning, paprika, salt, and black pepper.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken and sear for about 2–3 minutes per side, just until lightly golden. The chicken does not need to be fully cooked at this stage. Transfer to the prepared baking dish.
Reduce the heat to medium and add the butter to the same skillet. Add the thickly sliced mushrooms and cook for 5–6 minutes, stirring occasionally, until they begin to soften and develop some color. Add the garlic and cook for another 30 seconds.
Sprinkle the flour over the mushrooms and stir for about 1 minute. Gradually pour in the chicken broth while stirring to prevent lumps. Add the heavy cream, Italian seasoning, salt, and black pepper. Simmer gently for 3–4 minutes, until the sauce begins to thicken. It should remain slightly loose because it will continue thickening in the oven.
Pour the mushroom cream sauce around and over the chicken. Scatter the mozzarella and Parmesan over the casserole. Keep the cheese layer relatively light so some of the chicken and larger mushroom slices remain visible.
Bake uncovered for 20–25 minutes, or until the sauce is bubbling and the chicken reaches an internal temperature of 165°F (74°C) at its thickest point. For a lightly golden finish, broil for only 1–2 minutes at the end, watching carefully to prevent excessive browning. Allow the casserole to rest for 5 minutes before serving.